Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review Letters

Quantum Phase in Interferometry

Z. Hradil, Robert Myška, Jan Peřina, Michael Zawisky, Yuji Hasegawa, H. Rauch

Published Jun 3, 1996• 59 citations
Publisher

Abstract

Quantum estimation of a phase shift is formulated within quantum mechanics and information theory. Using Bayes' estimation, the phase can be attributed even to a single interfering particle. The method is demonstrated for neutron interferometry operating near the quantum limit.
DOI
10.1103/physrevlett.76.4295

Related papers

Phase estimation in interferometry

Michael Zawisky, Yuji Hasegawa, H. Rauch +3 more

An objective theory of phase shift estimation is formulated within quantum mechanics and quantum estimation theory. The general validity of Bayesian analysis even in the limit of very low particle numbers is demonstrated using the information of measured phase-sensitive data. The formalism was applied to the estimation of an unknown phase in neutron interferometry operating near the quantum limit, when only a few particles are registered, and also in the regime of large particle numbers.

Journal of Physics A Mathematical and General•Jan 16, 1998•29 citations

Optimal Quantum Measurements for Phase Estimation

Barry C. Sanders, Gerard J Milburn

Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic $1/n$ decrease in phase uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$ for $n$ the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the $1/n$ limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.

Physical Review Letters•Oct 16, 1995•242 citations

Optimal quantum measurements for phase-shift estimation in optical interferometry

Barry C. Sanders, Gerard J Milburn, Zhongxi Zhang

Quantum information theory, applied to optical interferometry, yields a 1/n scaling of phase uncertainty Δφ independent of the applied phase shift φ, where n is the number of photons in the interferometer. This 1/n scaling is achieved provided that the output state is subjected to an optimal phase measurement. We establish this scaling law for both passive (linear) and active (nonlinear) interferometers and identify the coefficient of proportionality. Whereas a highly nonclassical state is required to achieve optimal scaling for passive interferometry, a classical input state yields a 1/n scaling of phase uncertainty for active interferometry.

Journal of Modern Optics•Jul 1, 1997•19 citations

On the general problem of quantum phase estimation

D'Ariano, G M, Chiara Macchiavello, Sacchi, M F

The problem of estimating a generic phase-shift experienced by a quantum state is addressed for a generally degenerate phase shift operator. The optimal positive operator-valued measure is derived along with the optimal input state. Two relevant examples are analyzed: i) a multi-mode phase shift operator for multipath interferometry; ii) the two mode heterodyne phase detection.

•Jan 1, 1998•41 citations

Sub-Heisenberg phase uncertainties

Luca Pezzè

Phase shift estimation with uncertainty below the Heisenberg limit, $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}\ensuremath{\propto}1/{\overline{N}}_{T}$, where ${\overline{N}}_{T}$ is the total average number of particles employed, is a mirage of linear quantum interferometry. Recently, Rivas and Luis, [New J. Phys. 14, 093052 (2012)] proposed a scheme to achieve a phase uncertainty $\ensuremath{\Delta}\ensuremath{\phi}\ensuremath{\propto}1/{\overline{N}}_{T}^{k}$, with $k$ an arbitrary exponent. This sparked an intense debate in the literature which, ultimately, does not exclude the possibility to overcome $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}$ at specific phase values. Our numerical analysis of the Rivas and Luis proposal shows that sub-Heisenberg uncertainties are obtained only when the estimator is strongly biased. No violation of the Heisenberg limit is found after bias correction or when using a bias-free Bayesian analysis.

Physical Review A•Dec 12, 2013•23 citations

Quantum limits to lossy optical interferometry

Bruno M. Escher, R. L. de Matos Filho, L. Davidovich

An open question in optical interferometry is what is the fundamental quantum limit in the presence of photon losses. This question is addressed in the present paper, by finding an analytical lower bound for the uncertainty in the phase shift estimation, which implies, for any state of light, and even for arbitrarily small losses, that this uncertainty scales asymptotically with the number of photons at best as the shot noise.

Frontiers in Optics•Jan 1, 2010•0 citations
Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic 1/n1/n1/n decrease in phase uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$ for nnn the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the 1/n1/n1/n limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.
Quantum information theory, applied to optical interferometry, yields a 1/n scaling of phase uncertainty Δφ independent of the applied phase shift φ, where n is the number of photons in the interferometer. This 1/n scaling is achieved provided that the output state is subjected to an optimal phase measurement. We establish this scaling law for both passive (linear) and active (nonlinear) interferometers and identify the coefficient of proportionality. Whereas a highly nonclassical state is required to achieve optimal scaling for passive interferometry, a classical input state yields a 1/n scaling of phase uncertainty for active interferometry.
Phase shift estimation with uncertainty below the Heisenberg limit, $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}\ensuremath{\propto}1/{\overline{N}}_{T}$, where N‾T{\overline{N}}_{T}NT​ is the total average number of particles employed, is a mirage of linear quantum interferometry. Recently, Rivas and Luis, [New J. Phys. 14, 093052 (2012)] proposed a scheme to achieve a phase uncertainty $\ensuremath{\Delta}\ensuremath{\phi}\ensuremath{\propto}1/{\overline{N}}_{T}^{k}$, with kkk an arbitrary exponent. This sparked an intense debate in the literature which, ultimately, does not exclude the possibility to overcome $\ensuremath{\Delta}{\ensuremath{\phi}}_{\mathrm{HL}}$ at specific phase values. Our numerical analysis of the Rivas and Luis proposal shows that sub-Heisenberg uncertainties are obtained only when the estimator is strongly biased. No violation of the Heisenberg limit is found after bias correction or when using a bias-free Bayesian analysis.